History

…historical section eventually goes here..

…electricity and magnetism were discovered independently, Maxwell's equations in classical vector analysis which allows the formulation as a tensor FF as below, and “magnetism is a consequence of electrostatics and covariance, hence the composite noun electromagnetism”…

is constant. If one requires these constants all to be inside a discrete subgroup Γ↪ℝ\Gamma \hookrightarrow \mathbb{R}, then the data ({Ai},{λijmodΓ})(\{A_i\}, \{\lambda_{i j} mod \Gamma\}) defines a degree 2-cocycle in Cech-Deligne cohomology on XX with coefficients in ℝ/Γ\mathbb{R}/\Gamma. Below we see that experiment demands that such a subgroup exists and is given by the additive group of integers.

Kirchhoff’s laws

Kirchhoff's laws are a kind of coarse graining of Maxwell’s equations, where instead of infinitesimal quantities one considers actual macroscopic current and voltage?.

Therefore the above data is subject to the additional constraint that it induces well-defined U(1)U(1)-valued holonomy – this is Dirac’s quantization condition, a necessary requirement for the existence of quantum mechanical particles on XX that are charged under the background electromagnetic field.

Concretely: for any smooth curve γ:S1→X\gamma : S^1 \to X and any cover{Vi→S1}\{V_i \to S^1\} of S1S^1 refining the pullback of the cover UU to S1S^1, and for every triangulation {v,e}\{v, e\} of S1S^1 subordinate to {Ui→X}\{U_i \to X\}, i.e. such that there is an index map ρ\rho such that γ(e)⊂Uρ(e)\gamma(e) \subset U_{\rho(e)} and γ(v)⊂Uρ(v)\gamma(v) \subset U_{\rho(v)}

In short: the holonomy of the constant path on a point x∈Xx \in X must be 1∈U(1)1 \in U(1), but if that path sits in a triple intersection Ui∩Uj∩UkU_i \cap U_j \cap U_k then the holonomy is equivalently given as the exponentiated sum of the three transition functions. This forces the sum λij+λjk−λik\lambda_{i j} + \lambda_{j k} - \lambda_{i k} to land in ℤ↪ℝ\mathbb{Z} \hookrightarrow \mathbb{R}.

Traditionally physicist try to give that half-line a physical interpretation by imagining that it is the body of an idealized infinitely-thin and to one side infinitely-long solenoid. Indeed, such a solenoid would have a magnetic monopole charge on each of its ends, so if the one end is imagined to have disappeared to infinity, then the other one is the magnetic charge that Dirac imagines to sit at the origin of our setup.

In this context the half-line {x1≥0}\{x^1 \geq 0\} is called a Dirac string. While there is the possibility to sensibly discuss the idea that this Dirac string actually models a physical entity like an idealized solenoid, its main purpose historically is to confuse physics students and keep them from understanding the theory of fiber bundles. Therefore here we shall refrain from talking about Dirac strings and consider U:=ℝ3\{x1≥0}⊂XU := \mathbb{R}^3 \backslash \{x^1 \geq 0\} \subset X as exactly what it is, by itself: an open subset that is part of a cover of XX. Unfortunately, of course, Dirac didn’t mention the other open subsets in that cover (at least one more is needed for a decent discussion), so that the Dirac string keeps haunting physicists.

…running out of time…just quickly now…

…Dirac effectively considered the overlap cocycle condition Aj−Ai=somethingA_j - A_i = something, found that by the requirement that AA has well defined holonomy it follows that there must be gijg_{ij} a function with values in U(1)U(1) such that Aj−Ai=dloggijA_j - A_i = d log g_{ij}, then did away with the jj-patch (considering a kind of limit as we encircle the half axis) and concluded that AA must be the log-differential of a U(1)U(1)-valued function, whose winding number around the half-axis he identified with the magnetic charge, which in terms of bundles one identifies with the Chern-class of the bundle in question …

…have to run…

In modern terms:

The clutching construction gives U(1)U(1)-principal bundle ob S2S^2 by covering with two hemispheres U0U_0 and U1U_1 and picking a transition function g:S1→U(1)g \colon S^1 \to U(1) on the overlap U0∩U1≃S1×(0,ϵ)U_0 \cap U_1 \simeq S^1\times (0,\epsilon). The integral winding number of gg represents the first Chern class of the line bundle.

If we think (as we may) of U0=S2−{*}U_0 = S^2 - \{\ast\} as covering most of the 2-sphere except one point and of U1U_1 the ϵ\epsilon-open neighbourhood of that point, then this A0A_0 vanishes on most of the sphere and close to the point taken out (“the Dirac string”) it becomes non-vanishing and equal to g−1dgg^{-1}\mathbf{d}g.